Module

for

The Lorenz Attractor

     

Lorenz Attractor

    The Lorenz attractor is a set of differential equations which are popular in the field of Chaos. The equations describe the flow of fluid in a box which is heated along the bottom. This model was intended to simulate medium-scale atmospheric convection.  Lorenz simplified some of the Navier-Stokes equations in the area of fluid dynamics and obtained three ordinary differential equations

        [Graphics:Images/LorenzAttractorMod_gr_1.gif],  

        [Graphics:Images/LorenzAttractorMod_gr_2.gif]

        [Graphics:Images/LorenzAttractorMod_gr_3.gif].      
    
The parameter p is the Prandtl number, [Graphics:Images/LorenzAttractorMod_gr_4.gif] is the quotient of the Rayleigh number  and critical Rayleigh number  and b is a geometric factor.  Lorenz is attributed to using the values [Graphics:Images/LorenzAttractorMod_gr_5.gif].


    There are three critical points (0,0,0)  corresponds to no convection,  and the two points

     [Graphics:Images/LorenzAttractorMod_gr_6.gif] and  [Graphics:Images/LorenzAttractorMod_gr_7.gif]  correspond to steady convection.

The latter two points are to be stable, only if the following equation holds

    [Graphics:Images/LorenzAttractorMod_gr_8.gif].

Proof  Lorenz Attractor  Lorenz Attractor  

 

Computer Programs  Lorenz Attractor  Lorenz Attractor  

Program (Runge-Kutta Method in 3D space)  To compute a numerical approximation for the solution of the initial value problem  

    
[Graphics:Images/LorenzAttractorMod_gr_9.gif]  with  [Graphics:Images/LorenzAttractorMod_gr_10.gif],  
    
[Graphics:Images/LorenzAttractorMod_gr_11.gif]  with  [Graphics:Images/LorenzAttractorMod_gr_12.gif],  
    
[Graphics:Images/LorenzAttractorMod_gr_13.gif]  with  [Graphics:Images/LorenzAttractorMod_gr_14.gif],  

over the interval  
[Graphics:Images/LorenzAttractorMod_gr_15.gif]  at a discrete set of points.

Mathematica Subroutine (Runge-Kutta Method in n-dimensions).

[Graphics:Images/LorenzAttractorMod_gr_16.gif]

Example 1.  Solve the Lorenz I. V. P.  
        [Graphics:Images/LorenzAttractorMod_gr_17.gif]        and      [Graphics:Images/LorenzAttractorMod_gr_18.gif]  
        [Graphics:Images/LorenzAttractorMod_gr_19.gif]        and      [Graphics:Images/LorenzAttractorMod_gr_20.gif]  
    
    [Graphics:Images/LorenzAttractorMod_gr_21.gif]        and      [Graphics:Images/LorenzAttractorMod_gr_22.gif]  
Solution 1.

 

Example 2.  Solve the Lorenz I. V. P.  
        [Graphics:Images/LorenzAttractorMod_gr_78.gif]        and      [Graphics:Images/LorenzAttractorMod_gr_79.gif]  
        [Graphics:Images/LorenzAttractorMod_gr_80.gif]        and      [Graphics:Images/LorenzAttractorMod_gr_81.gif]  
    
    [Graphics:Images/LorenzAttractorMod_gr_82.gif]        and      [Graphics:Images/LorenzAttractorMod_gr_83.gif]  
Use Mathematica's NDSolve procedure.  
Solution 2.

 

Example 3.  Solve the Lorenz I. V. P.
        [Graphics:Images/LorenzAttractorMod_gr_112.gif]  
        [Graphics:Images/LorenzAttractorMod_gr_113.gif]  
    
    [Graphics:Images/LorenzAttractorMod_gr_114.gif]  
Solution 3.

 

Example 4.  Solve the Lorenz I. V. P.
        [Graphics:Images/LorenzAttractorMod_gr_183.gif]  
        [Graphics:Images/LorenzAttractorMod_gr_184.gif]  
    
    [Graphics:Images/LorenzAttractorMod_gr_185.gif]  
Use Mathematica's NDSolve procedure.  
Solution 4.

 

 

Rössler Attractor

    In 1976 the Swiss mathematician Otto Rössler was studying oscillations in chemical reactions and discovered another set of equations with attractor.  They also are involved with the study of  Navier-Stokes equations.  

        [Graphics:Images/LorenzAttractorMod_gr_240.gif],    
        
        [Graphics:Images/LorenzAttractorMod_gr_241.gif],       
        
        
[Graphics:Images/LorenzAttractorMod_gr_242.gif].       

Rössler is acclaimed to use the parameters where a = 0.2, b = 0.2, and c = 5.7.  Screw chaos occurs when the parameter values are a = 0.343,  b = 1.82  and  c = 9.75.  Rössler wanted to find a minimum system which would exhibit chaos. This system of equations looks easier than the Lorenz system, but it is harder to analyze.

 

Example 5.  Solve the Rössler attractor I. V. P.   
        [Graphics:Images/LorenzAttractorMod_gr_243.gif]          with  [Graphics:Images/LorenzAttractorMod_gr_244.gif],    
          [Graphics:Images/LorenzAttractorMod_gr_245.gif]          with  [Graphics:Images/LorenzAttractorMod_gr_246.gif],       
    
    [Graphics:Images/LorenzAttractorMod_gr_247.gif]      with  [Graphics:Images/LorenzAttractorMod_gr_248.gif].   
Solution 5.

 

Example 6.  Solve the Rössler attractor I. V. P.   
    [Graphics:Images/LorenzAttractorMod_gr_300.gif]              with  [Graphics:Images/LorenzAttractorMod_gr_301.gif],    
      [Graphics:Images/LorenzAttractorMod_gr_302.gif]          with  [Graphics:Images/LorenzAttractorMod_gr_303.gif],       
    [Graphics:Images/LorenzAttractorMod_gr_304.gif]      with  [Graphics:Images/LorenzAttractorMod_gr_305.gif].          
Solution 6.

 

Research Experience for Undergraduates

Lorenz Attractor  Lorenz Attractor  Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook Lorenz Attractor

 

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(c) John H. Mathews 2004